MaN)Random variable Review of probability and random variables Consider a binary communication system P(0)=0.30 ,0 Poi= P(receive 1 I sent 0)=0.01 Poo=P(receive 0 sent 0)=1-PoI=0.99 P(1)=0.71 1o= P(receive 0 I sent 1)=0.1 Pu= p(receive 1 I sent 1)=1-P10=0.9 What is the probability that the output of this channel is 1? Assuming that we have observed a 1 at the output, what is the probability that the input to the channel was a 1? Communications Engineering
Communications Engineering 11 Random variable Review of probability and random variables Consider a binary communication system
MaN)Random variable Review of probability and random variables A random variable is a mapping from the sample space Q2 to the set of real numbers a() Real Line Range Discrete r v: range is finite(0, 1) or countable infinite({0,12…} Continuous r v. range is uncountable infinite (real number) Communications Engineering
Communications Engineering 12 Random variable Review of probability and random variables ➢ A random variable is a mapping from the sample space to the set of real numbers ➢ Discrete r.v.: range is finite ({0,1}) or countable infinite ({0,1,…}) ➢ Continuous r.v.: range is uncountable infinite (real number)
MaN)Random variable Review of probability and random variables The cumulative distribution function(CDF)ofar.v. X IS F(x)=P(X≤x The key properties of cdf 1.0≤FA(x)≤ I with Fx(-∞)=0 and FX(∞)=1 2. Fux) is a non-decreasing function of X 3.F(x1<X≤x2)=F2(x2)-F2x) Communications Engineering 13
Communications Engineering 13 Random variable Review of probability and random variables ➢ The cumulative distribution function (CDF) of a r.v. X is ➢ The key properties of CDF
MaN)Random variable Review of probability and random variables The probability density function(PDf)of a r.V. X is fr(x)=Fx(x)or Fx(x)=L/(dy The key properties of PDF 1.Px(x)≥0 2. Px(x 3. P(x <X Sx2)=PX(x2)-P()=Px(x)dx (x) Area=P(x1<X≤x2) Area=f(x)dx P(x<Xsx,)f X X x2xxtar X Communications Engineering
Communications Engineering 14 Random variable Review of probability and random variables ➢ The probability density function (PDF) of a r.v. X is ➢ The key properties of PDF
MaN)Random variable Review of probability and random variables Common random variables. bernoulli. binomial Uniform, and gaussian Bernoulli distribution p(= P(X=a 0 1 Binomial distribution: the sum of n independent bernoulliv py(k) k , pR(1-pn-k where)=kI(n-k) Communications Engineering 15
Communications Engineering 15 Random variable Review of probability and random variables ➢ Common random variables: Bernoulli, Binomial, Uniform, and Gaussian ➢ Bernoulli distribution: ➢ Binomial distribution: the sum of n independent Bernoulli r.v